They're integrating the derivative of a function, so they used FTC. (The 1/2 coming out due to reverse chain rule basically.)

Note that (∂/∂v)(g(v+2t)) = g'(v+2t), where g' just means the derivative of the function g.

E.g. if g(y) was cos(y), we'd have ∂/∂v (g(v+2t)) = ∂/∂v (cos(v+2t)) = -sin(v+2t) (i.e. g' evaluated at v+2t, which is what g'(v+2t) means).

Then integrating -sin(v+2t), we'd get back to (1/2)cos(v+2t).

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